Understanding the statistics calculator
Enter numeric observations to calculate common descriptive statistics: mean, median, mode, minimum, maximum, range, variance, and standard deviation. These measures describe the dataset but do not by themselves establish causes or future predictions.
Mean, median and mode
The mean is the sum divided by the number of values. The median is the center after sorting. The mode is a value that occurs most often. In this calculator, if all numbers appear only once, the dataset has no mode; if several values tie for the highest frequency above one, all tied modes appear.
Population versus sample variation
Population variance divides the squared deviations by n. Sample variance divides them by n − 1, producing the commonly used unbiased estimator under its standard assumptions.
Population variance = Σ(x − mean)² ÷ n
Sample variance = Σ(x − mean)² ÷ (n − 1)
Standard deviation is the square root of the corresponding variance. It expresses spread in the original data units.
Worked example
Consider 2, 4, 4, 4, 5, 5, 7, 9. There are eight observations; the mean is 5, the median is 4.5, the mode is 4, and the range is 7. The population variance is 4, and the population standard deviation is 2.
When to use each summary
The mean takes all values into account but may be pulled by outliers. The median often describes a typical observation better for skewed data. Standard deviation is informative about spread, but interpreting it requires context about the distribution.
Input and precision notes
Use two to 500 finite numbers separated by commas, spaces or semicolons. Results are calculated at JavaScript numeric precision and displayed with up to five decimal places. Very large values can introduce floating-point rounding error.
Learn more
Read our mean, median, mode and standard deviation guide for a tutorial and sample/population decision guidance.